Download Greek Age, Worksheet 1 Early Greek Mathematics, including early

Greek Age, Worksheet 1
Early Greek Mathematics, including early proofs
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
1. Summarize (200-500 words) your textbook’s description of the mathematical view of the early
Greeks (Thales, Pythagoras, etc.)
2. Give geometric sketches (proofs) of the following identities:
(a) (a + b)2 = a2 + 2ab + b2
(b) (a + b)(a − b) = a2 − b2 .
(c) (a + b)3 = a3 + 3ab + 3ab2 + b3
(d) 1 + 3 + 5 + 7 + 9 + ... + 2n − 1 = n2
3. Expand out, algebraically, the expressions (a+b)4 and (a+b)5 . (Use the binomial theorem/Pascal’s
triangle to do this!) Then explain why the Greeks probably had a formula for (a + b)3 but not
(a + b)4 .
4. Write up, in your own words, Thales’ proof of the theorem on the triangle inscribed in a semicircle.
Then identify the various assumptions used in proving this result.
5. Prove Thales’ theorem on the bisection of a line segment: in order to bisect a line segment AB
draw two circles of equal radii centered at A and B respectively. Mark the points where these two
circles meet (let’s call these two new points C and D and draw the line segment overlineCD. The
intersection of CD and AB is the midpoint of AB.
(As you write up the proof, identify the arguments and assumptions used in proving this result!)
6. Find a proof of the Pythagorean Theorem and write it up in your own words.
(As you write up the proof, identify the arguments and assumptions used in proving the theorem.)
7. Identify the flaw in the false “proof” below. (In which line is the reasoning incorrect??)
Claim: cos x = sin x.
Proof.
2
cos x = sin x
(1)
2
(2)
cos x = sin x (Square both sides.)
2
2
sin x = cos x (Symmetric property of equality)
2
2
2
2
(3)
cos x + sin x = cos x + sin x (Add equals to equals.)
(4)
1 = 1 (Pythagorean theorem for trig functions)
(5)
Since 1 = 1 is true, our original identity is true!
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Greek Age, Worksheet 2
Triangular numbers, figurate numbers, sums of divisors
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
1. (Modification of Burton p. 103:4) Write the following as sums of three or fewer triangular numbers:
(a) 56
(b) 69
(c) 185
(d) 287
(e) 136
(f) 137
(g) The number of your first name
2. We denote by tn , the n-th triangular number tn := 1 + 2 + 3 + ... + n.
(a) Show geometrically that the n-th triangular number can be written in the form
tn =
n(n + 1)
.
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(b) Show Plutarch’s result that 8tn + 1 is a perfect square. Then give a geometric proof of this for
t3 by building a square out of 8 copies of t3 and one more point.
(c) Show that 9tn + 1 is a triangular number.
(d) (From Burton, p. 103: 5) For natural number n, show that (2n + 1)2 = (4tn + 1)2 − (4tn )2 .
3. The triangular number tn is the sum of the first n integers. Find formulae for the sums of the first
n squares and for the sums of the first n cubes. (If you find these formulae in a textbook or online,
please cite your source.)
4. In an earlier exercise you found the “number” of your first name. Compute the sum of the proper
factors of that number and decide if your number is deficient, perfect or abundant. Finally, take
the new number created by the sum of proper factors and see if that number is deficient, perfect or
abundant. (For example, the number associated with KEN is 20+5+50 = 75. The sum of proper
factors of 75 is 1+5+25+3+15= 49. So, sadly, I am deficient! The sum of proper factors of 49 is
1+7=8 so 49 is also deficient.)
5. (The sigma function.) Define the function σ : N → N by σ(n) is the sum of all factors of n. (Thus
σ(75) = (1 + 5 + 25 + 3 + 15) + 75 = 49 + 75 = 126.)
(a) What is σ(n) if n is perfect?
(b) Suppose n = 2α . Find a nice closed form for σ(n). (Hint: σ(n) is a geometric sum.)
(c) Suppose p is a prime and n = pα . Find a nice closed form for σ(n). (You should be able to
write σ(n) as a fraction.)
(d) Suppose p is a prime and n = 2α p. Find a simple closed form for σ(n).
(e) Suppose p = 2m − 1 is a prime and n = 2m−1 p. Find σ(n) and show that n is perfect.
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Greek Age, Worksheet 3
MATH 4367, Spring 2012
(Problem 1(c) requires an induction argument and 1(d) requires a limit. The remainder of this material does not
require mathematics beyond the high school curriculum.)
1. (From Burton p. 118:7) We form two coupled sequences {xn }, {yn } by defining
x1 = 2 and y1 = 3
and for n ≥ 2,
xn = 3xn−1 + 2yn−1 ,
yn = 4xn−1 + 3yn−1 .
(a) Write out the first five numbers in each sequence.
(b) Show that
2
yn2 − 2x2n = yn−1
− 2x2n−1 .
(c) Show that yn2 − 2x2n = y12 − 2x21 = 1.
(d) Show that the limit of the sequence
√
yn
is 2.
xn
2. Let x = φ represent the solution to the equation (1), below. Compute φ exactly.
x=1+
1
1+
(1)
1
x
Then show that φ also satisfies the following two equations:
x=1+
1
1+
and
x=1+
(2)
1
(1+
1
1+ 1
x
)
1
1+
(3)
1
(1+
1
1+
1
(1+
)
1 )
1+ 1
x
3. (Continuation of the previous problem.)
(a) We approximate the number φ from the previous problem by taking each of equations 1, 2
and 3, above and, on the right side of the equation, replacing x by the nice integer 1. What
fractions do we get when we do this? Are they close to φ?
(b) Instead of replacing x by 1 on the right side of the equations 1, 2 and 3, above, let us replace
x by 2. What do we get? Does this give a better approximation to φ?
4. The rectangle below has the property that the entire big rectangle (including both red and blue
parts) is similar to the smaller red rectangle. This means that the ratio of length to width of the
a+b
a
rectangle is
= . What is the ratio of length to width? (Find it exactly.)
a
b
3
(Hint: since we are talking about similar rectangles, we could make a = 1 and solve for 1 + b.)
5. Show that the following numbers are irrational:
√
(a) 3
(b) log10 (3)
(Hint: to show a number is irrational, begin by supposing instead that it is rational and in the form
M
where M and N are relatively prime integers. Then see if you can force a contradiction.)
N
6. Generalize the last problem by showing that
√
4
p and log10 (p) are irrational for any prime p.
Greek Age, Worksheet 4
Using the Euclidean Tools
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
Please cite any sources you use. Please take time to briefly explain your work or computations.
(Mathematics always involves explanation! ,)
1. Let us say that the line at right has length one:
Sketch, according to the Euclidean tools, line segments of length
(a)
(b)
(c)
(d)
(e)
(f)
4
3
2
5
4
x where x is the length of the segment at right:
3
√
5
√
4
5
√
1+ 5
φ :=
2
(Please provide commentary and explanations with your sketches.)
2. In the figure below, the lengths of segments CD, BD
√ and BA are all 1. The lengths of segments
1+ 5
AC and AD are equal to the golden ratio φ :=
. From the previous problem, we know that
2
the golden raio is a constructible length so the figure below is constructible.
(I will explicitly describe the construction: on a line mark a point A and from A, mark B and C
of lengths 1 and φ from A. Then at A draw a circle of radius φ and at B draw a circle of radius 1.
Those circles will intersect at D. With a straightedge, draw the line segment CD, BD, AD.)
D
C
B
A
=⇒ Determine all the angles in the figure above.
(Hint: note that the golden ratio φ has the property that
in this figure!)
1
φ
= φ−1 and so there are similar triangles
3. The Greeks knew how to inscribe certain regular polygons inside a circle. Suppose we are give a
circle of radius 1 (a “unit circle”.) Describe the construction of an inscribed regular polygon with
(a) six sides
5
(b) twelve sides
(c) five sides
(d) ten sides
(Hint for part (c): see the previous problem.)
4. Describe in depth Theodorus’s method of constructing
23.)
6
√ √
√
2, 3, ..., 17. (This is Burton p. 120, #
Greek Age, Worksheet 5
Using the Euclidean Tools
MATH 4367, Spring 2012
(Problem 1 requires Taylor series from a second semester calculus class. None of the remaining material requires
mathematics beyond the high school curriculum.)
Please cite any sources you use. Please take time to briefly explain your work or computations.
√
1. (a) Use√the√Taylor polynomial
of degree 2 for f (x) = a2 + x to write out best approximations
√
for 2, 3 and 5.
√
2
(b) Find the Taylor
polynomial
√ √
√ of degree 3 for f (x) = a + x and use it write out best approximations for 2, 3 and 5.
√
√
(c) Modern computers have been used to find approximations for π, 2 and 3. How accurate (to
how many digits) are those approximations today?
2. Use DeMoivre’s magical formula ((cos θ + i sin θ)n = cos nθ + i sin nθ) to create an expression for
(a) cos 2x as a polynomial in cos x. (Get rid of any occurrences of sin x by using the Pythagorean
theorem.)
(b) sin 2x in terms of sin x. (This won’t be a polynomial; it will involve a square root.)
(c) cos 3x as a polynomial in cos x.
(d) cos 4x as a polynomial in cos x.
3. Let x = cos 20o . Use the formula for cos 3x to obtain a cubic polynomial in x for which cos 20o is a
root.
4. In an earlier problem you found formulae for cos 2x and sin 2x in terms of cos x and sin x. Turn
these formula into “half-angle formulae“ by replacing 2x by θ and x by θ/2 and then solving for
cos θ and sin θ.
Then find the exact values of cosine and sine of the following angles.
(a)
(b)
(c)
(d)
θ
θ
θ
θ
= sixty degrees (θ = π/3 radians)
= 30o (= π/6 radians)
= 15o (= π/12)
= 7.5o (= π/24)
5. Suppose you are given the angle θ in the picture below, and so you know the length AD is the
length of OA multiplied by sin θ.
A
O
D
θ
C
B
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(a) Find the length of AC in terms of OA and θ. (Hint: use the half-angle formula for the sine
function from an earlier excercise.)
(b) Find the circumferences of regular polygons inscribed in the unit circle assuming the polygon
has
i. six sides
ii. 12 sides
iii. 24 sides
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Greek Age, Worksheet 6
Pythagorean triples
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
1. (a) Consider the two families of Pythagorean triples given in the notes:
i. (2n + 1, 2n2 + 2n, 2n2 + 2n + 1)
ii. (2n, n2 − 1, n2 + 1)
Show that these families fit the earlier description of all PPTs (u2 − v 2 , 2uv, u2 + v 2 ) by
providing u and v for each family.
(b) Find all primitive Pythagorean triples (a, b, c) in which the integers a, b, c are all less than 100.
(This is a continuation of an earlier exercise.)
(c) How many of these PPTs with values less than 100 are not described by the two families of
Pythagorean triples given above?
2. For each value of c ∈ {5, 9, 13, 17, 25, 29, 37, 41} construct a PPT (a, b, c).
3. Continue problem 1 by finding all PPTs (a, b, c) where c < 100.
4. Prove the first part of Euclid’s result on Primitive Pythagorean Triples (Theorem 1 in my notes.)
Assume that the Pythagorean Triple (a = u2 − v 2 , b = 2uv, c = u2 + v 2 ) is primitive (that is, no
positive integer but 1 divides all three terms) and show that this implies both that GCD(u, v) = 1
and that exactly one of the integers u, v is even.
5. Prove the second part of Theorem 1: assume that p > 2 is a prime dividing the three integers
a = u2 − v 2 , b = 2uv, and c = u2 + v 2 . Show that p divides both u and v.
6. (From Burton p. 117: 5 or Eves p. 97: 3.6 (g).)
(a) Show that if (x, x + 1, z) is a Pythagorean triple then so is (a, a + 1, c) where a = 3x + 2z + 1
and c = 4x + 3z + 2. Conclude then that there are an infinite number of Pythagorean triples
where the short sides are have integer length and differ by just 1.
(b) Find at least five Pythagorean triples where the short sides are integers which differ by just 1.
√
(c) Use your answers in part (b), above, to find five rational approximations to 2.
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Greek Age, Worksheet 7
Some Diophantine Equations
MATH 4367, Spring 2012
(None of this material requires mathematics beyond the high school curriculum.)
1. (from Eves, 197:6.15(a)) Use the method of false position to solve the following problem:
1
“Diophantus passed 16 of his life in childhood, 12
in youth and 17 more as a bachelor. Five years
after his marriage was born a son who died four years before his fater, at 12 his father’s final age.”
How old was Diophantus when he died?
2. Solve the following problem by the method of false position: (Eves, p. 240, problem 7.4 (7).) A
merchant pays duty on goods at three different places as he travels. At the first stop he must pay
one-third of the value of his goods. At the next he pays one-quarter of the remainder left over from
the previous tax. At the third place he pays one-fifth of what remains of his goods. His total duty
paid is 24. What was the original value of his goods?
3. (from Eves, 197:6.15(b)) Solve problem 17 in Book I of Diophantus’ Arithmetica: Find four numbers
where the sum of every arrangement three at a time is 22, 24, 27, 20.
4. (from Eves, 197:6.15(c)) Solve problem 16 in Book VI of Diophantus’ Arithmetica: A right triangle
ABC has a right angle at C. The point D is on the segment BC so that AD bisects the angle A.
Find the smallest integers for AB, AD, AC, BD, DC such that the ratios DC : CA : AD = 3 : 4 : 5.
5. (from Eves, 197:6.15(d)) The mathematician De Morgan who lived in the XIXth Century, apparently upon reading Diophantus, suggested the following problem. ”I was x years old in the year x2 .
When was I born?”
6. (Eves 197: 6.16 (a)) Show that
(a2 + b2 )(c2 + d2 ) = (ac ± bd)2 + (ad ∓ bc)2 .
Then use this identity to write 481(= 13 · 37) as a sum to two squares in two different ways.
7. (Eves 197: 6.16 (b)) Write 1105 = (5)(13)(17) as a sum of two squares in four different ways.
8. (Eves 197: 6.16 (c)) Suppose two integers, m and n differ by 1. Suppose also that x, y, a are integers
such that
x + a = m2 and y + a = n2 .
Show that xy + a is a perfect square.
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Greek Age, Worksheet 8
The Euclidean Algorithm
MATH 4367, Spring 2012
On all problems below, the instructions “Use the Euclidean Algorithm...” assume that all computations
done with the Euclidean Algorithm are done in the tabular format.
1. Finish the following table, to show that 521 and 641 are relatively prime. Then use your work to
find the the integers s and t such that 1 = 641s + 521t.
−q
-1
-4
-2
as + bt
a = 641
b = 521
120
41
s
1
0
1
-4
t
0
1
-1
5
1
0
2. Use the Euclidean algorithm in tabular format to find the GCD of 108 and 605 and compute the
integers s and t such that GCD(108, 605) = 108s + 605t.
3. Compute the inverse of 108 in Z605 .
4. Use the Euclidean algorithm to find the integer x such that 1 = 200x + 641y. (The integer x is “the
inverse of 200 mod 641.”)
5. Find the “inverse” of 521 modulo 625. (That is, find an integer s such that 521s ≡ 1 mod 625.)
6. Use the Euclidean algorithm to find the GCD of a = 232 + 1 and b = 214 · 52 − 1 and compute
the integers s and t such that GCD(a, b) = as + bt. (Fermat, probably around 1640, claimed that
232 + 1 is prime. Was he correct?)
7. Find the smallest pair of integers a and b such that the Euclidean Algorithm, in tabular format,
has nine rows? More generally, describe the “worse case” scenario for the Euclidean algorithm? In
particular what would be the smallest pair of numbers that would require 20 rows? 1000 rows? n
rows?
8. Use Bezout’s Identity to prove Euclid’s Lemma: If a prime p divides ab then either p divides a or
p divides b. (Hint: consider the contrapositive. If p doesn’t divide a and p doesn’t divide b then p
is relatively prime to a and relatively prime to b. Can you get from there, via Bezout’s Identity, to
the claim that p is relatively prime to ab?)
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Greek Age, Worksheet 9
What is the center of a triangle?
MATH 4367, Spring 2012
We examine a triangle from the modern viewpoint, placing it in the Cartesian plane and coordinatizing
it. Let’s place the vertex A at the origin (0, 0) and the vertex B at the point (b, 0) on the x-axis. The
third vertex C will have a general coordinate (c, d). We speak of this triangle as having vertices A(0, 0),
B(b, 0), C(c, d) and sides BC, AC and AB.
1. Find the equations for the three lines bounding the triangle.
2. (The centroid)
(a) Find the coordinates for the midpoints of each of the three sides.
(b) Find the three equations for the lines from a vertex through the midpoint of the opposite side.
Put those equations in standard (Ax + By = C) form.
(c) Show that the three lines in part (c) meet at a point. Give the coordinates of that point.
3. (Circumcenter) We continue to work with the general triangle with vertices A(0, 0), B(b, 0) and
C(c, d).
(a) Find the equation for each of the perpendicular bisectors of the three sides. Write these
equations in standard form.
(b) Show that these three lines meet in a point. What is the coordinate of that point?
4. Describe (in less than a page) how Euclid would have drawn the lines in the previous problem.
Draw pictures to illustrate your description.
5. (Incenter)
(a) For each vertex, A, B and C, draw the line which bisects that vertex.
(b) Give the equations of the three lines in part (a). Put the equations in standard form.
(c) Show that these three lines meet in a point. What is the coordinate of that point?
6. Describe (in less than a page) how Euclid would have drawn the lines in the previous problem.
Draw pictures to illustrate your description.
7. (Orthocenter) An altitude of a triangle is a line through a vertex perpendicular to the opposite side.
(a) For each vertex, A, B and C draw the altitude from that vertex through the opposite side.
(This is called “dropping a perpendicular” from the vertex.)
(b) Give the equations of the three lines in part (a) and put those equations in standard form.
(c) Show that these three lines meet in a point. What is the coordinate of that point?
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